Dynamic Mode Decomposition of inertial particle caustics in Taylor-Green flow
Omstavan Samant, Jaya Kumar Alageshan, Sarveshwar Sharma, Animesh Kuley
DDynamic Mode Decomposition ofinertial particle caustics in Taylor-Green flow
Omstavan Samant , Jaya Kumar Alageshan , Sarveshwar Sharma , and AnimeshKuley Centre for Fusion, Space and Astrophysics, University of Warwick, Coventry, CV4 7AL, UK. Department of Physics, Indian Institute of Science, Bangalore, India - 560012. Institute For Plasma Research, Gandhinagar, Gujarat, India - 382428. * [email protected] ABSTRACT
Inertial particles advected by a background flow can show complex structures. We consider inertial particles in a 2D Taylor-Green (TG) flow and characterize particle dynamics as a function of the particle’s Stokes number using dynamic modedecomposition (DMD) method from particle image velocimetry (PIV) like-data. We observe the formation of caustic structuresand analyze them using DMD to (a) determine the Stokes number of the particles, and (b) estimate the particle Stokes numbercomposition. Our analysis in this idealized flow will provide useful insight to analyze inertial particles in more complex orturbulent flows. We propose that the DMD technique can be used to perform similar analysis on an experimental system.
Advection of particles, such as dust or aerosol by a background flow is a ubiquitous phenomena. And the study of dispersionof these inertial particles are of immense interest both for applied and natural processes, in particular, to analyze oil spills inoceans , dispersion of pollutants and toxic elements , suspended particles in aquatic systems , formation of clouds andvolcanic plumes , and the effect of the flow patterns generated by the breathing action and cough on the dispersion of theaerosol particles are crucial to understand the spread of COVID-19 virus . Numerical studies of inertial particle dispersionin different types of flows, ranging from static to turbulent flows have shown that the particles display complex dynamicalbehaviours like formation of fractal clusters and caustics . The analysis of the structures formed by the particles encodeinformation about the Stokes number of the particles and the flow patterns.Experimental techniques such as particle image velocimetry (PIV) have been used to track particles and extract velocityprofiles when it is possible to identify individual particles in an image, but not with certainty to track it between images. Ifthe particle concentration is so low that it is possible to follow an individual particle it is called particle tracking velocimetry(PTV). While similar techniques have been adopted to track particles in simulations, the averaging process reduces the spatialresolution, which is critical in our application. In simulations, the Osiptov’s method have been used to extract causticfeatures , which track each particle in PTV like situations but fail for PIV like data. We propose the use dynamical modedecomposition (DMD) based scheme to obtain the spatio-temporal particle distributions as a representative for particle density.DMD methods have been used to extract coherent structures in simulations and experiments of fluids. We use the DMDmethod to analyze and extract the features of the caustics to (a) determine the Stokes number of the particles, and (b) estimatethe relative particle concentrations in a bi-disperse Stokes number system. Our approach can also be extended to multipleStokes number poly-disperse systems.This paper is organized as follows. In Sec. 1.1 we present the form of the TG flow, minimal model of an inertial particle,and the relevant numerical simulation details. In Sec. 1.2 we show the formation of caustic structures and analyze them usingDMD method in Sec. 1.3 and demonstrate how we extract the caustic wavefronts from the DMD mode. We use the positionand the gradient of the wavefront in the DMD eigen mode to estimate the Stokes number and the composition of a bi-disperseStokes number systems in Sec. 2 and present our conclusions in Sec. 3. We consider a 2D lattice of vortices in the form of a Taylor-Green (TG) flow. The TG flow is a steady state solution to theforced, incompressible Navier-Stokes equation and can be considered a convection model in 2D . Such a flow can be a r X i v : . [ phy s i c s . f l u - dyn ] F e b xperimentally setup using ion solutions in an array of magnets . The TG flow is given by the vorticity field as ω ( x , y ) = ω sin (cid:18) π xL (cid:19) sin (cid:18) π yL (cid:19) (1)and the corresponding velocity field is u ( x , y ) = V sin (cid:0) π xL (cid:1) cos (cid:16) π yL (cid:17) − cos (cid:0) π xL (cid:1) sin (cid:16) π yL (cid:17) (2)where x , y ∈ [ , L ) and are periodic, u is the Eulerian velocity field such that ω = ∇ × u . We choose V as the velocity scale and L as the length scale and write the system parameters in corresponding dimensionless form. We model the aerosol particles assmall rigid spheres, which are effectively points, that have density different from the surrounding fluid. The equation of motionof the inertial particles in a background flow given by the simplified Maxey-Riley approximation for small particles that aremuch denser than the fluid are d x dt = v d v dt = St ( u − v ) (3)where St is the Stokes number which captures the effect of particle inertia, x is the particle position and v is the particle velocity(see Appendix A for validity of the equations). The case when St → v = u . We use RK4 numerical scheme to discretize and evolve the particle positions and velocities.Furthermore, we use periodic boundary conditions, such that the particles are reintroduced into the system when they exit theboundary. In our analysis we set the time step to ∆ t = . ( L / V ) . A typical feature of inertial particles in a background flow is that they are expelled from high vorticity regions. In a backgroundflow with vortices, the inertial particles can form caustics . So in the case of TG flow, the inertial particles tend to accumulatealong the low vorticity regions that separate the vortices. We simulate the dynamics of inertial particles in a TG flow andobserve spatial regions where the Lagrangian velocity field is multi-valued, which are referred to as caustics (see Fig. 3(a)).Figure. 1 shows the caustic structure formed by a mono-disperse inertial particle system, when we start the simulation withthe particle positions initialized with uniform random distribution within the L × L box, and setting the initial velocities of theparticles to zero. In the long time limit, the particles accumulate along a curve . We observe that the caustics that form in thetransient state are robust and stable to small perturbations (see Appendix A), whereas the steady state structures break up andlead to chaos . Furthermore, the steady state behaviour strongly depends on the system size and the boundary conditions.We find that for a range of Stokes numbers the caustic structures preserve their shapes; and their sizes depend on St . In thefollowing section we use dynamic mode decomposition (DMD) to extract features of the structures, in particular the causticwavefront, and study its size dependence on the Stokes number. Furthermore, the sharpness of the caustic wavefront enables usto detect and extract their sizes even in presence of poly-disperse Stokes number systems. The caustics in Fig. 1 have a complex structure and in the presence of multiple Stokes number particles resolving these structuresfrom a single snap-shot is hard. Therefore we employ the spatio-temporal data in the form of a video sequence that contains F frames of N × N pixel images and analyze them using the dynamic mode decomposition (DMD) method.DMD is a data analysis technique that has been used to extract coherent structures in fluid dynamic systems , where itis able to extract different modes that are similar to normal modes in linear dynamical systems. The DMD is a data-driventechnique introduced by Schmid as a numerical procedure for extracting dynamical features from flow data . The DMDalgorithm takes in a time-series data in the form of vectors { (cid:126) v ,(cid:126) v , ...(cid:126) v T } and estimates a linear dynamical system that cangenerate a map (cid:126) v i + = A (cid:126) v i (4)where A is an N × N matrix and the eigenvectors of A form the DMD modes, with the corresponding eigenvalues. Finally,the eigenvectors are reshaped into N × N pixel image to obtain the modes. A Singular Value Decomposition (SVD) basedalgorithm for estimating the DMD modes is described in Appendix B.Let i stand for the iteration number such that the particles are in their stationary initial state and start their evolution at i = v i are obtained by rearranging the N × N pixel images at instant i into N × igure 1. A snapshot of the particle distribution in (a) at t = ∆ t shows the complex structure formed by themono-dispersed inertial particles with St = . L × L . The background color-map corresponds tothe vorticity field and the particles are plotted using black markers. Notice that the particles in the high vorticity regions havemoved out towards the regions bounding the vortices. The zoomed-in version in (b) shows the details of the caustic structuresaround the central region of the domain. See the video C.1 for the evolution of the caustic structures.analysis we employ { v , v , ... v } (i.e. F = D ( α ) ( j , k ) represent the ( j , k ) th pixel of the α th eigenmode, ordered in terms of decreasing absolute eigenvalue. Since the caustics arelocalized around the central region of the domain, we use a zoomed-in region of size 512 ×
512 pixels (i.e. N = D ( ) , shown in Fig. 2(a) highlights astraight-line caustic structure, which we refer to as the wavefront. The eigenvalues of other DMD modes decay exponentially.We observe that the position of the wavefront in D ( ) has a systematic dependence on the Stokes number, and to extract thisrelation we detect the location of the wavefronts using edge detection techniques. Similarly, when we perform DMD analysison a bi-disperse system D ( ) ( j , k ) shows two distinct wavefronts (see Fig. 2(b)) corresponding to the two different Stokesnumbers; and here DMD uses the velocity information to unambiguously extract the wave fronts. In particular, Fig. 3(a) showsthe reduced phase space portrait of a typical particle which form the caustic and Fig. 3(b) shows the particles overlaid on topof the DMD that demonstrates the DMD’s ability to extract the caustic structures. Furthermore, we find that the intensity ofeach wavefront compared to the background, which we refer to as prominence , depends on the corresponding initial particleconcentrations in the system.We now prescribe a method to extract the position of the wavefront from DMD. We use a Sobel operator such that the igure 2. The highest singular DMD eigenvector, D ( ) , obtained for: (a) St = . St = { , } with 7 : 3 ratio of initial particle concentrations. The horizontal lines correspond to the causticwavefronts, and the number of such fronts indicate the different Stokes number particles. Also notice that the wavefrontscorresponding to St = . Figure 3. (a) The particle trajectory in the reduced phase space, ( y , v y ) , shows the multi-valued nature of the casutics invelocity between the dotted vertical lines . The movie in C.2 shows the evolution of the bi-disperse particles overlaid on thecorresponding DMD from Fig. 2(b) and the plot (b) shows a snap-shot at the 650 th iteration step when the caustics and theDMD wavefronts coincide. Notice that the first DMD picks out only the slow moving horizontal caustics.vertical gradient of the first DMD mode is given by G ( j , k ) = ∂ D ( ) ( j , k ) ∂ y ≈ ∆ y − − − (cid:126) D ( ) ( j , k ) (5)where (cid:126) represents the 2D convolution operator , and ∆ y is the spacing in DMD along the y -axis. We then sum over the valuesin the x-direction to get a 1D function of y as (cid:104) G (cid:105) x = (cid:90) L G ( j , k ) dx ≈ N ∑ j = G ( j , k ) ∆ x (6)where ∆ x is the spacing in DMD along the x -axis, and we choose a square grid with ∆ x = ∆ y such that (cid:104) G (cid:105) x is by definitionindependent of the grid spacing and is dimensionless. In the next section we describe how the location and the value of thepeaks in (cid:104) G (cid:105) x can be used to find the Stokes number of the particles and the relative initial concentrations in the case of abi-disperse system. Results
In (a) we show the plots of (cid:104) G (cid:105) x obtained for different values of the Stokes number from mono-disperse systems.We extract the location of the peaks in (cid:104) G (cid:105) x for each St , as defined in eq. 7, to generate the plot in (b) and we find that the Y WF and St have a quadratic dependence.The (cid:104) G (cid:105) x is obtained from the DMD as described in Sec. 1.3 by simulating mono-disperse Stokes number particle systemsto generate the plots in Fig. 4(a) which shows (cid:104) G (cid:105) x as a function of St . The alignment of the peaks in (cid:104) G (cid:105) x along a curveindicates a systematic dependence of the location of wavefront on the Stokes number. To extract this relation we first get thelocation of the caustic wavefront from the domain center using the position of the peaks in (cid:104) G (cid:105) x given by Y WF = arg max y (cid:104) G (cid:105) x − L y gives the value of y for which (cid:104) G (cid:105) x is maximum. We then plot Y WF as a function of St as shown in Fig. 4(b).Using a non-linear least squares fit method we find that the relation is of the form Y WF ∼ a St + b St + c , with values ofthe parameters a , b , and c as indicated in Fig. 4(b), where x represents St . Now, extrapolating the fit we find that Y WF = St = . St > . St of new particle systems. In particular we demonstrate that the above method can be generalized to work in caseof bi-disperse system.As shown in Fig. 2(b), for a bi-disperse system the DMD has two sets of caustic wavefronts, corresponding to each Stokesnumber. Now we set one of the Stokes number fixed at one ( St = . St and find (cid:104) G (cid:105) x to generate the plots in Fig. 5.The results in Fig. 5 show that even in the bi-disperse system the caustic wavefront has the same characteristic behaviour on theStokes number as the mono-disperse system. In particular, the wavefront corresponding to St has a fixed location and thewavefront due to St preserves the dependence on Y WF of the mono-disperse system. Our studies with poly-disperse St systemsshow that the caustic wavefronts can be used to find the Stokes number of different particles in the system using the relationobtained from a mono-disperse system. Until now the bi-disperse systems that we considered had equal number of St and St particles, with uniform initial distributionin space. Now we study the variation in D ( ) w.r.t. the change in relative number of particles. We observe that the intensityof the wavefront or the gradient in the DMD image depends on the number of particles or the initial uniform concentration,denoted by C ( St ) .The variable (cid:104) G (cid:105) x gives the gradient of D ( ) along the vertical direction and the magnitude of the gradient indicatessharpness of the wavefronts (see Fig. 6(a)). To measure the sharpness of the wavefront we define a "prominence" parameter, P , as the sum of the non zero values of ¯ |(cid:104) G (cid:105) x | in the neighbourhood of the wavefront, which takes into account multiplepeaks in the vicinity of the wavefront. We find that the prominence of the wavefront has a systematic dependence on the initialconcentration of the corresponding Stokes number particles and from Fig. 6(b) we find that on a log-log plot the relation is igure 5. Plots of (cid:104) G (cid:105) x for a bi-disperse system, with one of the Stokes number fixed at one ( St = . St .Notice that the peaks in (cid:104) G (cid:105) x corresponding to St are aligned at the same location along y , whereas the peaks due to St showsimilar trend as the plots for mono-disperse systems in Fig. 4(a). Figure 6.
The plot (a) shows the variation in the absolute value of (cid:104) G (cid:105) x for a bi-disperse system, with St = St = C ( St ) / ( C ( St ) + C ( St )) . Notice that the peaks corresponding to eachwavefront is not unique and have a finite spread in y. In (b) the relation between the prominence corresponding to St and St are given by P , P respectively, as a function of the initial particle concentrations C is shown in a log-log plot. The linear fitshows that the ratios of the peaks of ˆ |(cid:104) G (cid:105)| and the ratios of the concentration are related by a power-law, with a power close to-1.linear, with a slope approximately equal to -1. This implies that in a bi-disperse system the ratio of the prominence is inverselyrelated to the ratio of initial concentrations. We can use this relation to predict the concentration of various Stokes numberparticles in a system. We study the dynamics of inertial particles in a Taylor-Green flow with periodic boundary conditions in 2D. In a minimalmodel of inertial particles we observe that for a mono-disperse Stokes number system, starting from a uniform distribution ofstationary particles, the particle distribution forms caustics in the strain dominated region of the flow. We use the DMD methodto analyze the PIV-like time-series data of the spatio-temporal particle distribution and find that the largest absolute eigenvaluemode is effective in extracting the caustic wavefront-like structure. We notice that (a) the position of the wavefront dependson the particle Stokes number and employ standard image processing techniques to quantitatively extract a quadratic relation.Using this relation we can predict the Stokes number from the wavefront position. Furthermore, we find that for a bi-dispersesystem the DMD is able to extract two different wavefronts corresponding to each Stokes number and the positions of eachwavefront follow the same quadratic relation as in the case of mono-disperse system. We also observe that (b) the sharpnessof the wavefront in the DMD, measured in terms of prominence, depends on the initial particle concentration and find thatfor a bi-disperse Stokes number system the ratio of the wavefront prominence is inversely proportional to the corresponding tokes number initial concentration. Hence the measurement of prominence can be used to estimate the concentration of thecorresponding Stokes number particles.We propose that the DMD technique can be used to analyze real experimental PIV data of caustics and perform similaranalysis to extract information about the Stokes numbers and concentrations of the particles. In future we will consider detailedNavier-Stokes equation for the self-consistent evolution of the velocities and analyze the caustic structures in turbulent flows.
A Validity of particle dynamics
The particle dynamics we use in our study is a special case of d x dt = v d v dt = St ( u − v ) + R (cid:18) u . ∇ u + v . ∇ u (cid:19) + η ( v , t ) (8)where η is the white noise and R = ρ f / ( ρ p + ρ f / ) for density of particle ρ p and density of fluid ρ f . We present our resultsfor η = R =
0, and use small values of these parameters to verify the stability of our results.
B DMD algorithm
The N × N pixel image at k th instant, I k ( i , j ) , is rearranged into an N × X k ( m ) (note that we subtract the mean valuefrom X k ( m ) ). Now the F sequence of image frames are appended together to form N × N T matrix Y . Let the N × ( F − ) dimensional matrix formed from the first ( F − ) frames be Y and the last ( F − ) frames be Y , i.e. (cid:104) I k (cid:105) N × N −→ (cid:104) X k (cid:105) N × (cid:2) X | X | ... | X N T (cid:3) −→ [ Y ] N × N T (cid:104) X | X | ... | X ( N T − ) (cid:105) −→ [ Y ] N × ( N T − ) (cid:2) X | X | ... | X N T (cid:3) −→ [ Y ] N × ( N T − ) (9)We find the singular value decomposition (SVD) of the matrix Y , such that Y = U Σ V ∗ , where U is a N × N complexunitary matrix, Σ is an N × ( F − ) rectangular diagonal matrix with non-negative real numbers on the diagonal, and V is a ( F − ) × ( F − ) real or complex unitary matrix.Now choose a lower dimensional SVD matrices made up of first n T ( << F ) columns, represented by ˜ U , ˜ V , and the first n T × n T block of Σ as ˜ Σ . Define a matrix n T × n T matrix A as A = ˜ U ∗ Y ˜ V ˜ Σ − (10)Then the dynamic modes are the N × A that is rearranged into N × N matrix. C Movies
The movies referred to in the text are available in the additional materials.
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Acknowledgements
This work is supported by Board of Research in Nuclear Sciences (BRNS Sanctioned no. 39/14/05/2018-BRNS), Scienceand Engineering Research Board EMEQ program (SERB Sanctioned no. EEQ/2017/000164), Infosys Foundation YoungInvestigator Award, and IISc startup grant. We thank Rahul Pandit for valuable inputs. JKA acknowledges useful discussionswith Akhilesh Kumar Verma.
Author contributions statement
J.K.A, S.S., and A.K. conceived the problem. O.S. and J.K.A. performed simulations. All authors analysed the results. O.S.and J.K.A. wrote the main manuscript text. All authors reviewed the manuscript.
Additional information
Corresponding author
Correspondence to Jaya Kumar Alageshan ([email protected]).
Competing interests